Much but not all of the notation used here is similar
or identical to the one used in the standard textbook on Kolmogorov
complexity by Li and Vitányi [30].

Since sentences
over any finite alphabet are encodable as bitstrings, without loss of
generality we focus on the binary alphabet .
denotes the empty string,
the set of finite sequences over ,
the set of infinite sequences over ,
.
stand for strings in .
If then is the concatenation of and (e.g.,
if and then
).
Let us order lexicographically: if
precedes alphabetically (like in the example above)
then we write or ; if
may also equal then we write or
(e.g.,
).
The context will make clear where we also identify with
a unique nonnegative integer
(e.g., string 0100 is represented by integer 10100 in the dyadic
system or
in the decimal system).
Indices
range over the positive integers,
constants over the positive reals,
denote functions mapping integers to integers,
the logarithm with basis 2,
for real .
For
,
stands for the real number with dyadic expansion
(note that
for ,
although
).
For , denotes the number of bits in ,
where
for
;
.
is the prefix of consisting of
the first bits, if ,
and otherwise (
).
For those that contain at least one 0-bit,
denotes the lexicographically smallest
satisfying
( is undefined for of the form ).
We write if there
exists such that
for all .

For notational simplicity we will use the sign also to indicate
summation over uncountably many strings in , rather than
using traditional measure notation and signs. The reader should
not feel uncomfortable with this notational liberty -- density-like
nonzero sums over uncountably many bitstrings, each with individual
measure zero, will not play any critical role in the proofs.